PDHonline Course E331 (3 PDH)
Fundamentals of PID Control
Instructor: Anthony K. Ho, PE
2020
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Table of Content
1. History and Background of PID Control .............................................................1
2. Theory of PID Control .........................................................................................2
2.1. Proportional Control..................................................................................3
2.2. Integral Control .........................................................................................5
2.3. Derivative Control .....................................................................................6
3. Implementation of PID Control ...........................................................................9
3.1. Choosing the structure of a PID controller ...............................................9
3.2. Tuning the PID controller .......................................................................11
4. Open-loop vs. Closed-loop Control ...................................................................13
5. Example - PID Controller for DC Motor ...........................................................14
6. Cascade Control .................................................................................................17
7. Troubleshooting PID Control ............................................................................19
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1. History and Background of PID Control
The original technology for industrial proportional, integral, and derivative (PID)
control was pneumatic, hydraulic, or mechanical and the controller usually had a
simple interface for manual tuning. The first theoretical analysis of a PID
controller dates back to 1922 when the Russian American engineer Nicolas
Minorsky developed an automatic ship steering system for the U.S. Navy based on
observations of the steersmen’s use of current error, past error, and rate of change
to keep the ship on course. Controllers with electrical systems were developed
after World War II.
PID control is used to control and maintain processes. It can be used to control
physical variables such as temperature, pressure, flow rate, and tank level. The
technique is widely used in today’s manufacturing industry to achieve accurate
process control under different process conditions. PID is simply an equation that
the controller uses to evaluate the controlled variables. A process variable (PV)
temperature, for example, is measured, and a feedback signal is sent to the
controller. The controller then compares the feedback signal to the set point (SP)
and generates an error value. The value is examined with one or more of the three
proportional, integral, and derivative methodologies. As a result, the controller
issues the necessary commands or alters the control variable (CV) to correct the
error (E). These procedures form an iterative process. Below is a common control
loop application.
In this example, oil is flowing into the tank in a non-constant rate. The oil level in
the tank is the process variable, which is measured, and this information is fed
back to the controller. An operator enters a set point for a desired level. The
controller compares the current level with the set point and generates a value that is
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examined with a PID method. The controller then adjusts the valve position to
correct the error.
2. Theory of PID Control
PID controllers typically use control loop feedback in industrial and control
systems applications. The controller first computes a value of error as the
difference between a measured process variable and a preferred set point. It then
tries to minimize the error by increasing or decreasing the control inputs or outputs
in the process so that the process variable moves closer to the set point. This
method is most useful when a mathematical model of the process or control is too
complicated or unknown for the system. To increase performance, such as by
increasing the system’s responsiveness, PID parameters must be adjusted
according to the specific application.
The block diagram below shows an example of a heating process that is controlled
by a PID controller (programmable logic controller or PLC in this case). The
temperature of the furnace is controlled by adjusting the gas valve. The operator
sets the desired temperature as the set point. The furnace’s temperature is
measured, and this information is sent back to the controller. The feedback is
compared to the set point, and an error value is calculated. The PID equations then
determine the suitable valve position to correct the error. In fact, this is an example
of a PID feedback control loop.
The following figure shows a typical step response curve after a controller
responds to a set point change. The curve rises from 10% to 90% of final steady
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state value within a period known as the rise time. The curve rises from 0% to
63.2% of peak value within a period known as the step response time. The rise
time is equal to the step response time minus the dead time.
One of the advantages of PID is that for many processes there are straightforward
correlations between the process responses and the use and tuning of the three
terms (P, I, and D) by the controller. Designing a PID system involves two steps.
First, the engineer must choose the structure of the PID controller, for example P
only, P and I, or all three terms P, I, and D. Second, to tune the controller, the
engineer must choose numerical values for the PID parameters.
These three parameters for the PID algorithm are the proportional, integral, and
derivative constants. The proportional constant determines the reaction based on
the current error; the integral constant determines the reaction according to the
total of recent errors; and the derivative constant determines the reaction using the
rate at which the errors have been changing. These three actions are then used to
adjust the process through control elements such as the position of a valve. In
simple terms, P depends on the current error, I depends on the sum of past errors,
and D predicts future errors based on the current rate of change of errors.
2.1. Proportional Control
The proportional element of PID examines the magnitude of the error, and the PID
control reacts proportionally. A large error receives a large response. For example,
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if there is a large temperature error, the fuel valve would be opened a lot, while a
small error would receive a small response. In mathematical terms, the
proportional term (Pout) is expressed as:
Pout = Kp e
where
Pout: Proportional portion of controller output
Kp: Proportional gain
e:
Error signal, e = set point – process variable
e = SP – PV here represents a reverse acting loop. When e = PV – SP, it refers to a
direct-acting loop. In a direct-acting loop, the process variable is greater than the
set point; therefore, the appropriate controller action is to increase the output. A
typical example of a direct-acting system is one that controls temperature using
cooling water. In a reverse-acting loop, the process variable is less than the set
point; therefore, the controller’s action is to decrease the output. An example
would be a system that controls temperature using steam.
The following figure illustrates a proportional control and shows that there is
always a steady state error in proportional control. The error will decrease with
increasing proportional gain, but the tendency toward oscillation will also increase.
Let’s look at a furnace as an example. Assume that between 1500°F and 2000°F,
the system response is proportional such that it opens the fuel valve in proportion
to the error. When the temperature is below 1500°F, the valve is open 100%. When
it is above 2000°F, the valve is closed or 0% open. The proportional band (also
known as the reciprocal of gain) in this case is assumed to be 500°F. However, the
proportional band is usually expressed in percentage. The percentage is calculated
by dividing the proportional band in the engineering unit (°F) by the full controller
range and multiplying by 100. The full controller range in this case is 2000°F –
70°F = 1930°F (Assume 70°F is the room temperature.) Therefore, the
proportional band in percentage is equal to (500°F / 1930°F) × 100 or 26%. The
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engineer can adjust the proportional band and make the system more or less
responsive to an error. The rule of thumb is that a smaller the proportional band (or
large gain) results in a large output or faster response to a given input error, and a
larger band (or small gain) results in a less responsive controller. However, one
must be aware that an excessively large proportional gain will lead to process
instability and oscillation.
One may see that there are issues with proportional control only. One of these
issues is that proportional control cannot compensate for very small errors. (These
errors are also known as offset.) Another issue is that it cannot adjust its output
based on the rate of change in the measured variable. For example, if you are
driving on the highway at 55 miles per hour and the car in front of you slows
down, you may respond by applying 50% brake pressure. This is a proportional
response, because the car in front is slowing down but not coming to a stop.
However, if the car slows down even more, you will apply more brake pressure,
since the car in front’s rate of stopping is greater. In other words, we apply more
brake pressure as we see the distance between our cars getting smaller, because we
naturally respond to the rate of change of error. However, proportional control
systems do not do this; they only respond to the magnitude of the error, not to its
rate of change.
2.2. Integral Control
To address the first issue with the proportional control, integral control attempts to
correct a small error (offset). The integral examines the error over time and
increases the importance of even a small error over time. The integral is equal to
error multiplied by the time the error has persisted. A small error at time zero has
zero importance. A small error at time 10 has an importance of 10 times error. As
such, the integral increases the response of the system to a given error over time
until it is corrected. The integral can also be adjusted, and the adjustment is called
the reset rate. The reset rate is a time factor. The shorter the reset rate, the quicker
the correction of an error. However, too short a reset rate can cause erratic
performance. In hardware-based systems, the adjustment can be done by a
potentiometer that changes the time constant of an RC circuit. Most of today’s
applications use a software-based control, such as a PLC module, through which
the engineer changes the parameter of the reset rate. The mathematical expression
of an integral-only controller (Iout) is:
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where
Iout:
Ti:
Ki:
e:
Integral portion of controller output
Integral time, or reset time
Integral gain
Error signal, e = set point – process variable
2.3. Derivative Control
The derivative part of the control output attempts to look at the rate of change in
the error signal. The derivative will cause a greater system response to a rapid rate
of change than to a small rate of change. In other words, if a system’s error
continues to rise, the controller must not be responding with sufficient correction.
The derivative senses this rate of change in the error and provides a greater
response. The derivative is adjusted as a time factor and therefore is also called rate
time. It is essential that too much derivative should not be applied or it can cause
overshoot or erratic control. In mathematical terms, the derivative term (Dout) is
expressed as:
where
Dout:
Td:
Kd:
e:
Derivative portion of controller output
Derivative time
Derivative gain
Error signal, e = set point – process variable
To summarize the tasks of all three controls, proportional control causes an input
signal to change as a direct ratio of the error signal variation. It responds
immediately to the current tracking error, but it cannot achieve the desired set point
accuracy without an unacceptably large gain. Thus, the proportional term usually
needs the other terms. Integral control causes the output signal to change as a
function of the integral of the error signal over time duration. The integral term
yields zero steady state error in tracking a constant set point. It also rejects constant
disturbances. Derivative action reduces transient errors and causes an output signal
to change as a function of the rate of change of the error signal. The contributions
of the three terms will yield the control output, or the control variable:
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Control Variable = Pout + Iout + Dout
The following figure illustrates a closed-loop system with proportional, integral,
and derivative control. (Closed-loop system will be defined later.)
In practice, most PID controllers can be run in two modes: manual or automatic. In
manual mode, the controller output is manipulated directly by the operator,
typically by pushing buttons that increase or decrease the controller output. A
controller may also operate in combination with other controllers, such as in a
cascade or ratio connection, or with nonlinear elements, such as multipliers and
selectors. In automatic mode, the PID parameters can be adjusted during operation.
When there are changes of modes and parameters, it is important to avoid
switching transients.
Since the PID controller is a dynamic system, it is necessary to make sure the state
of the system is correct when switching the controller between manual and
automatic mode. When the system is in automatic mode, the PID algorithm
produces a control variable that may be different from the manually generated
control variable. Therefore, it is crucial for the operator to make sure the two
control variables match at the time of switching. This is called bumpless transfer.
The first scenario is switching from automatic to manual. A bumpless transfer will
force the manual output to track the output of the controller. If no disturbances
occur after switching, the PV will not change. The second scenario is switching
from manual to automatic. If the PV is not equal to the SP, the switch will result in
an immediate “bump” to the controller output due to proportional action. To avoid
the immediate bump, the operator can do one of the following: 1) adjust the SP to
the current PV and then switch to automatic and slowly change the SP; 2) slowly
adjust the controller output until the PV equals the current SP and then switch to
automatic; or 3) use SP tracking while in manual. This would force the current SP
to track the PV while in manual so that no proportional bump occurs while
switching to automatic.
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Below is the pseudo-code of a simple software loop that implements a feedback
PID algorithm.
Constants:
Gain
ResetRate
Derivative
Variables:
Input
InputDev
InputPrev
Error
Setpt
OutputTemp
Output
Feedback
Mode
Action
System gain
Reset rate
Derivative
Process input
Process input plus derivative
Process input from previous pass, used in derivative calculation
Difference between input and set point
Set point
Temporary value of output
Output of PID algorithm
Result of time delay in positive feedback loop (Sometimes this
is called the manual reset or bias)
Value is “AUTO” if loop is in automatic
Value is “DIRECT” if loop is direct acting
IF Mode = “AUTO” THEN
InputDev = Input + (Input – InputPrev) * Derivative * 60 // Derivative
InputPrev = Input
Error = Setpt – InputDev
// Error based on reverse action
IF Action = “DIRECT” THEN
Error = 0 – Error
// Change sign of error for direct action
ENDIF
OutputTemp = Error * Gain + Feedback
// Limit output to 0 – 100%
IF OutPutTemp > 100 THEN OutPutTemp = 100
IF OutPutTemp < 0 THEN OutPutTemp = 0
Output = OutputTemp
// The final output of the controller.
Feedback = Feedback + (Output – Feedback) * ResetRate / 60
ELSEIF Mode = “MANUAL” THEN
InputPrev = Input
// Stay ready for bumpless switch to Auto
Feedback = Output
ENDIF
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Feedback control corrects any cumulative measurement errors or other errors in the
system. In contrast, feedforward control applies to a system in which a balance
between supply and demand is achieved by measuring both demand potential and
demand load and using this information to govern supply. Feedforward control
applies to a process that must be completely understood and requires no special
instrumentation. Feedforward is often used to control processes with transportation
and measurement lags. For example, a feedforward value representing “cold water
poured into a warm mix” could boost the output value faster than waiting for the
process variable to change as a result of the mixing. Another example is that the
lag between input and output in a distillation column may vary from a few minutes
to many minutes, depending on the size of the column and other variables. When
long lags are inherent to this process, feedback control becomes difficult, because
it cannot make any required corrections until the error is sensed. Feedforward can
be used to reduce the lag and improve control results, because an exact balance
between supply and demand potential is achieved before the process is influenced.
3. Implementation of PID Control
As suggested earlier, to implement a PID control, the engineer must first choose
the structure of the PID controller and then must choose numerical values for the
PID coefficients to tune the controller.
3.1. Choosing the Structure of a PID Controller
To choose the structure of a controller, refer to the following table of tuning effects
of the PID controller terms. This is a reference table that shows how each term of
the controller can be selected to accomplish a particular closed-loop system effect
of specification.
P
Reference Tracking Tuning
Disturbance Rejection Tuning
Step Reference
Constant Load Disturbance
Transient
Increasing Kp > 0
speeds up the
response.
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Steady State
Increasing Kp > 0
reduces but does
not eliminate
steady state
offset.
Transient
Increasing Kp > 0
speeds up the
response.
Steady State
Increasing Kp > 0
reduces but does
not eliminate
steady state
offset.
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I
D
Introducing
integral action Ki
> 0 gives a wide
range of
response types.
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Introducing
integral action Ki
> 0 eliminates
offset in the
reference
response.
Derivative action Derivative action
Kd > 0 gives a
has no effect on
wide range of
steady state
responses and
offset.
can be used to
tune response
damping.
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Introducing
integral action Ki
> 0 gives a wide
range of
response types.
Introducing
integral action Ki
> 0 eliminates
steady state
offsets.
Derivative action
Kd > 0 gives a
wide range of
responses and
can be used to
tune response
damping.
Derivative action
has no effect on
steady state
offset.
Let’s consider this example: if it is necessary to remove steady state offsets from a
closed-loop output response, the table indicates that a D-term will not do this and a
P-term will reduce the offset if the proportional gain Kp is increased, but an I-term
will eliminate the offset completely. Therefore, an integral term would be chosen
for inclusion in the controller structure. Another example occurs when the engineer
has elected to use integral action to eliminate constant steady state process
disturbance offset errors but now wishes to speed up the closed-loop system
response. The table shows that increasing the proportional gain Kp will have just
this effect, so both proportional (P) action and integral (I) action would be selected,
giving a PI controller solution.
The examples above show how the PID controller structure is selected to match the
desired closed-loop performance. This selection process can be simplified using
the following PID term selection flow diagram.
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3.2. Tuning the PID Controller
The second part of setting up a PID controller is to tune or choose numerical values
for the PID parameters. PID controllers are tuned in terms of their P, I, and D
terms. Tuning the control gains can result in the following improvements of
responses:
Proportional gain (Kp)
Larger proportional gain typically means faster response, because the larger the
error, the larger the proportional term compensation. However, an excessively
large proportional gain may result in process instability and oscillation.
Integral gain (Ki)
Larger integral gain implies that steady state errors are eliminated faster.
However, the trade-off may be a larger overshoot, because any negative error
integrated during transient response must be integrated away by positive error
before steady state can be reached.
Derivative gain (Kd)
Larger derivative gain decreases overshoot but slows down transient response
and may lead to instability due to signal noise amplification in the
differentiation of the error.
The following table lists some common tuning methods and their advantages and
disadvantages. The choice of method will mostly depend on whether or not the
loop can be taken offline for tuning and the response time of the system. If the
system can be taken offline, the best tuning method often involves subjecting the
system to a step change in input, measuring the output as a function of time, and
using this response to determine the control parameters. Manual tuning methods
can be quite inefficient, especially if the loops have response times of longer than a
minute.
Tuning
Method
Manual tuning
ZieglerNichols
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Advantages
No math required
Online method
Proven method
Online method
Disadvantages
Requires experienced
personnel
Process upset
Some trial and error
Very aggressive tuning
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Software tools
Consistent tuning
Either online or offline tuning
May include actuator and
sensors analysis
Good process models
Cohen-Coon
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Involves some cost and
training
Requires some math
Offline method only
Only good for first-order
process
Many manufacturing and industrial process companies have in-house guidelines
for tuning PID controllers in particular process plant units. Therefore, it is often
possible to provide rules and empirical formulas for PID controller tuning
procedures. Some of these guidelines base their procedures on the routines of the
commonly used Ziegler-Nichols methods. The two Ziegler-Nichols methods use an
online process experiment followed by a set of rules to calculate the numerical
values of the PID parameters. Numerous improvements and extensions of the
associated rules have been achieved since their introduction.
Ziegler-Nichols is a type of continuous cycling method for controller tuning. The
term continuous cycling refers to a continuous oscillation with constant amplitude
and is based on trial-and-error procedure. The following are the steps to implement
the method:
1. Allow the process to reach steady state as much as possible, turn off the integral
mode (or set time to zero), and then turn off the derivative mode.
2. Assign a small value to proportional-only controller gain K (e.g., 0.5), and place
the controller in automatic mode.
3. Make a small set point change so that the control variable moves away from the
set point.
4. Increase the gain slightly.
5. Repeat steps 3 and 4 until continuous oscillation is achieved. This is known as
the ultimate gain.
6. Calculate the PID controller settings using the Ziegler-Nichols tuning relations
in the following table.
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7. Evaluate the Ziegler-Nichols controller settings by introducing a small set point
change and observing the closed-loop response. Fine tune the settings if
necessary.
Kp
P Control
PI Control
PID Control
0.5 K
0.45 K
0.6 K
Ki
Kd
Note: K is the ultimate gain (the gain at which the oscillations continue with
constant amplitude), and T is the time or ultimate period in minutes measured in
the gain calibration above (the time it takes the process variable to complete one
full oscillation while the system is at steady state).
By tuning the three parameters in the PID controller algorithm, the controller can
provide control action designed for specific process requirements. The controller’s
response can be described in terms of the controller’s responsiveness to error, the
degree to which the controller overshoots the set point, and the degree of system
oscillation. Note that the use of the PID algorithm for control does not guarantee
optimal control of the system or system stability. Some applications may require
using only one or two modes to provide the appropriate system control. A PID
controller will be called a P, I, PI, or PD controller in the absence of respective
control actions. If the PID controller parameters are improperly chosen, the
controlled process input can become unstable. Instability is often caused by excess
gain, particularly in the presence of significant time delay. In most applications,
stability of response is required, and the process must not oscillate for any
combination of process conditions.
4. Open-Loop vs. Closed-Loop Control
An open-loop controller, also called a non-feedback controller, computes its input
into a system using only the current state and its model of the system. The
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controller does not receive a feedback signal from the process and it only has a setpoint and a fixed output signal. The output signal does not change, regardless of
the system conditions and disturbances. Consequently, an open-loop system cannot
engage in machine learning or correct any error that it produces. It also may not
compensate for disturbances in the system.
For example, an irrigation sprinkler system programmed to turn on at fixed times is
an example of an open-loop system if it does not measure soil moisture as a form
of feedback. Even when it is raining, the sprinkler system would still activate on
schedule.
An open-loop controller is often used in simple processes because of its simplicity
and low cost, particularly in systems where feedback is not critical. A typical
example would be a conventional dryer, for which the length of machine drying
time is entirely based on the judgment and estimation of the human operator. To
obtain a more accurate or more adaptive control, it is necessary to feed the output
of the system back to the controller. This type of system is called a closed-loop
system.
In a closed-loop system, also called a feedback system, the controller has a
feedback signal from the process. The controller has a set point, a feedback input
signal, and a varying output signal. The output signal increases or decreases
proportionally to the error of the set point compared to the input signal. The input
signal varies proportionally to the system disturbances and the gain of the
measurement sensor.
An example of a closed-loop system would be an automobile’s cruise control.
When the car goes up a hill, the car will power up to maintain the set point speed
set by the driver; the steeper the hill, the more power will be applied. The increase
in slope is a system disturbance, but there can be more than one disturbance on a
system. A stronger head wind would add to the error of an increasing slope,
requiring the car use even more power to keep up with the set point.
5. Example: PID Controller for DC Motor
In this example, the desired target speed of the motor is set by the user. This value
is fed into the speed controller to change the motor speed, and the loop is closed by
a tachometer. The controller constantly adjusts the value of the DC voltage applied
to the motor to maintain the desired speed. The control loop is shown in the
following figure:
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The speed controller contains two components: speed control and a digital-toanalog converter. The speed control component compares the desired speed with
the measured speed and generates a digital value proportional to the DC value to
be applied to the motor. The digital-to-analog converter is implemented using a
signal generator and a low pass filter to smooth out the signal.
The transfer function for the DC motor’s speed is expressed in Laplace form as:
where
K:
R:
L:
J:
b:
V:
θ:
Electromotive force control = 0.01 Nm/Amp
Electrical resistance = 1 Ω
Electrical inductance = 0.5H
Moment of inertia of rotor = 0.01 kg m2/s2
Damping ratio of mechanical system = 0.1 Nms
Source voltage
Rotating speed
Therefore, the rotating speed of the motor is directly proportional to the input
voltage. First, let’s try a proportional-only control with a gain (Kp) of 100 and step
input of 1 rad/sec. A step response is received as follows:
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From the figure above, we see that both the steady state error and overshoot are too
large. Since adding an integral term will eliminate the steady state error and a
derivative term will reduce the overshoot, we try a PID controller with small Ki = 1
and Kd = 1. After that, the step response curve looks like this:
However, now the settling time is too long. So we increase Ki to 200 to reduce the
settling time. After that, the step response curve looks like the following:
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Now we see that the response is much faster than before; however, the large Ki has
worsened the transient response (big overshoot). So we increase Kd to 10 to reduce
the overshoot. After that, the step response curve looks like:
We can now use a PID controller with the following parameters to adjust the value
of the DC voltage applied to the motor to maintain the desired speed.
Kp = 100
Ki = 200
Kd = 10
6. Cascade Control
PID control can be improved by changing the PID parameters, improving
measurement, such as sampling rate, precision, and accuracy, and using low-pass
filtering. Another proven method is cascading multiple PID controllers, which can
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be used to yield better dynamic performance. Using a previous example, a cascade
control would look like this:
In a cascade control, there are two PID controllers, with one controlling the set
point of the other. A PID controller acts as an outer loop (primary) controller,
which controls the primary physical parameter, such as temperature. The other
controller then acts as an inner loop (secondary) controller, which reads the output
of the outer loop controller as its set point, usually controlling a more rapid
changing parameter, such as flow rate. The inner loop must have a response time
that is faster than that of the outer loop. Since the outer loop controller provides the
set point for the inner loop, the inner loop must be faster than the outer loop so that
the set point will not change too fast for the inner loop to catch; otherwise, stability
problems may result. Below is a block diagram of a cascade control loop.
Cascade control is especially advantageous when the main disturbance is in
the inner loop. This is because with the cascade configuration, the correction of
the inner disturbance occurs as soon as the secondary sensor detects the upset.
There are three main advantages gained by the use of cascade control:
1. Disturbances affecting the secondary variable can be corrected by the secondary
controller before their influence is felt by the primary variable.
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2. Closing the control loop around the secondary part of the process reduces the
phase lag seen by the primary controller, resulting in increased speed of the
response. Because of the increased speed of response, the sensitivity of the
primary process variable to process upsets is also reduced.
3. The use of the secondary loop can reduce the effect of control valve sticking or
actuator nonlinearity.
7. Troubleshooting PID Control
If a control loop is not performing satisfactorily, troubleshooting is necessary to
identify the source of the problem. In many process applications, a control loop
that once operated satisfactorily can become either unstable or excessively sluggish
for a variety of reasons, including:
Changing process conditions, usually changes in throughput rate
A sticking control valve stem
A plugged line in a pressure or differential pressure transmitter
Fouled heat exchangers, particularly reboilers for distillation columns
Cavitating pumps (usually caused by a suction pressure that is too low)
The starting point for troubleshooting is to obtain enough background information
to define the problem clearly. The following questions must be answered:
What are the control objectives?
What is the process being controlled?
What are the control variables?
Are closed-loop response data available?
Is the controller in manual or automatic mode? Is it reverse or direct acting?
If the process is cycling, what is the cycling frequency?
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What PID controller algorithm is used? What are the PID controller
parameters?
Is the process open-loop stable?
What additional documentation is available, such as control loop summary
sheets, piping and instrumentation diagrams (P&ID), etc.?
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